The Jacobi triple product is a mathematical identity discovered by Carl Gustav Jacob Jacobi and published in 1829 in his book on the theory of elliptic functions, Fundamenta Nova Theoriae Functionum Ellipticarum. It states that an infinite product built from terms of the form (1 minus x to the 2m), (1 plus x to the 2m-1 times y squared), and (1 plus x to the 2m-1 divided by y squared) equals an infinite sum of terms of the form x to the n squared times y to the 2n, for complex numbers x and y with the absolute value of x less than 1 and y not equal to zero. The identity has deep connections elsewhere in mathematics: it is the Macdonald identity for the affine root system of type A1, and it is the Weyl denominator formula for the corresponding affine Kac-Moody algebra. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Origin Yearintroduced by Jacobi (1829) in his work Fundamenta Nova Theoriae Functionum Ellipticarum Connections
Attributed To
Source Jacobi Triple Product (Wikipedia)
In Branch
Source Wolfram MathWorld: Jacobi Triple Product
Is Kind Of Object
Sources
1. Jacobi Triple Product (Wikipedia)
Lead section
In mathematics, the Jacobi triple product is the identity: ∏ m = 1 ∞ ( 1 − x 2 m ) ( 1 + x 2 m − 1 y 2 ) ( 1 + x 2 m − 1 y 2 ) = ∑ n = − ∞ ∞ x n 2 y 2 n , {\displaystyle \prod _{m=1}^{\infty }\left(1-x^{2m}\right)\left(1+x^{2m-1}y^{2}\right)\left(1+{\frac {x^{2m-1}}{y^{2}}}\right)=\sum _{n=-\infty }^{\infty }x^{n^{2}}y^{2n},} for complex numbers x and y, with |x| < 1 and y ≠ 0.
Attributed To: Carl Gustav Jacob Jacobi, Lead paragraph
It was introduced by Jacobi (1829) in his work Fundamenta Nova Theoriae Functionum Ellipticarum.
View the SourceWolfram MathWorld: Jacobi Triple Product
Wolfram MathWorldIn Branch: Analysis, Subject classificationsQuote, In Branch: Analysis, Subject classifications
Calculus and Analysis Special Functions
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.